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Strong Shape Invariance of Alexander-Spanier Normal Homology Theory

Anzor Beridze

Abstract

In the paper [Ba-Be-Mdz], using the Alexander-Spanier cochains based on the normal coverings, the exact homology theory $\bar{H}^N_*(-,-;G)$, the so called Alexander-Spanier homology theory, is defined. In the paper we will use the method of construction of the strong homology theory to show that the homology theory $\bar{H}^N_*(-,-;G)$ is a strong shape invariant.

Strong Shape Invariance of Alexander-Spanier Normal Homology Theory

Abstract

In the paper [Ba-Be-Mdz], using the Alexander-Spanier cochains based on the normal coverings, the exact homology theory , the so called Alexander-Spanier homology theory, is defined. In the paper we will use the method of construction of the strong homology theory to show that the homology theory is a strong shape invariant.
Paper Structure (2 sections, 10 theorems, 75 equations)

This paper contains 2 sections, 10 theorems, 75 equations.

Table of Contents

  1. Homology theory of pro-chain complexes
  2. A new strong shape invariant homology theory

Key Result

Lemma 1

For each direct system ${\bf C}^*=\left\{C^*_\lambda, p_{\lambda \lambda'}, \Lambda\right\}$ of cochain complexes and an injective abelian group $I$, the homomorphisms are isomorphisms, where ${\bf C}_*=\left\{C^\lambda_*, p_{\lambda \lambda'}, \Lambda\right\}$ is the inverse system of chain complexes $C^\lambda_*=C_*^\lambda \left(\beta_\#\right)$.

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 7 more